Forward–backward splitting in bilaterally bounded Alexandrov spaces
针对带尖边的几何曲面等非黎曼流形上的优化问题，提出并证明了在曲率上下有界的Alexandrov空间中前向后向方法的收敛性，并在三维简单几何曲面上进行了数值验证。
Abstract With the goal of solving optimisation problems on non-Riemannian manifolds, such as geometrical surfaces with sharp edges, we develop and prove the convergence of a forward–backward method in Alexandrov spaces with curvature bounded both from above and from below. This bilateral boundedness is crucial for the availability of both the gradient and proximal steps, instead of just one or the other. We numerically demonstrate the behaviour of the proposed method on simple geometrical surfaces in $$\mathbb {R}^3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> </mml:math> .